Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction

Sheikh, Nadeem Ahmad; Ali, Farhad; Saqib, Muhammad; Khan, İlyas; Jan, Syed Aftab Alam; Alshomrani, Ali Saleh; Alghamdi, Metib

doi:10.1016/j.rinp.2017.01.025

Cited by 198 publications

(90 citation statements)

References 27 publications

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“…In another study, Morales-Delgado et al [23] compared the solutions obtained by using the CF derivative and the Liouville-Caputo derivative. Sheikh et al [24] used the analysis of the Atangana-Baleanu (AB) and CF for generalized Casson fluid model. Atangana and Alkahtani [20] modeled the groundwater flowing within a confined aquifer by using the CF derivative.…”

Section: Introduction and Some Preliminariesmentioning

confidence: 99%

European Vanilla Option Pricing Model of Fractional Order without Singular Kernel

2018

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Abstract:Recently, fractional differential equations (FDEs) have attracted much more attention in modeling real-life problems. Since most FDEs do not have exact solutions, numerical solution methods are used commonly. Therefore, in this study, we have demonstrated a novel approximate-analytical solution method, which is called the Laplace homotopy analysis method (LHAM) using the Caputo-Fabrizio (CF) fractional derivative operator. The recommended method is obtained by combining Laplace transform (LT) and the homotopy analysis method (HAM). We have used the fractional operator suggested by Caputo and Fabrizio in 2015 based on the exponential kernel. We have considered the LHAM with this derivative in order to obtain the solutions of the fractional Black-Scholes equations (FBSEs) with the initial conditions. In addition to this, the convergence and stability analysis of the model have been constructed. According to the results of this study, it can be concluded that the LHAM in the sense of the CF fractional derivative is an effective and accurate method, which is computable in the series easily in a short time.

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Section: Introduction and Some Preliminariesmentioning

confidence: 99%

European Vanilla Option Pricing Model of Fractional Order without Singular Kernel

2018

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“…These definitions include mainly the Riemann-Liouville and the Liouville-Caputo fractional-order derivatives. [19][20][21][22][23][24][25] Due to FC is one of the most powerful mathematical tools used in the recent decades to model real-world problems, several existing electrical circuits models have been generalized, and fractional derivatives models have been developed to represent the behavior of fractional linear electrical systems, for the designing of analog and digital filters, as well as to describe the magnetically coupled coils or the behavior of circuits and systems with memristors, meminductors, or memcapacitors. 15 For the Riemann-Liouville operator, the derivative of a constant is not zero, and the Laplace transform involves terms without physical signification.…”

Section: Introductionmentioning

confidence: 99%

“…A lot of studies have been done about this operator with excellent results. [19][20][21][22][23][24][25] Due to FC is one of the most powerful mathematical tools used in the recent decades to model real-world problems, several existing electrical circuits models have been generalized, and fractional derivatives models have been developed to represent the behavior of fractional linear electrical systems, for the designing of analog and digital filters, as well as to describe the magnetically coupled coils or the behavior of circuits and systems with memristors, meminductors, or memcapacitors. In other studies, [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] authors developed electrical circuits models using FC; they used the Riemann-Liouville or Liouville-Caputo fractional-order derivative operators.…”

Section: Introductionmentioning

confidence: 99%

Fractional operator without singular kernel: Applications to linear electrical circuits

Morales‐Delgado

Gómez-Aguilar

Taneco-Hernández

et al. 2018

Circuit Theory & Apps

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This paper presents the analytical solutions of fractional linear electrical systems by using the Caputo-Fabrizio fractional-order operator in Liouville-Caputo sense. This novel operator involves an exponential kernel without singularities. The fractional equations were solved analytically by using the properties of Laplace transform operator, as well as the convolution theorem. To validate the analytical solutions, numerical simulations were carried out. KEYWORDSCaputo-Fabrizio fractional operator, exponential decay law kernel, fractional linear electrical circuits, Laplace transform 2394

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“…However, some issues were pointed out against both derivatives, including one in the Caputo sense and one in the Riemann-Liouville sense. As Sheikh [48] pointed out, the CF fractional derivative as the kernel in integral was nonsingular but was still nonlocal. Some researchers also concluded that the operator was not a derivative with fractional order but a filter with fractional parameter.…”

Section: Introductionmentioning

confidence: 99%

Optimality conditions for fractional differential inclusions with nonsingular Mittag–Leffler kernel

Bahaa

Hamiaz

2018

Adv Differ Equ

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In this paper, by using the Dubovitskii-Milyutin theorem, we consider a differential inclusions problem with fractional-time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. The Atangana-Baleanu fractional derivative of order α in the sense of Caputo with respect to time t, is considered. Existence and uniqueness of solution are proved by means of the Lions-Stampacchia theorem. The existence of solution is obtained for all values of the fractional parameter α ∈ (0, 1). Moreover, by applying control theory to the fractional differential inclusions problem, we obtain an optimality system which has also a unique solution. The controllability of the fractional Dirichlet problem is studied. Some examples are analyzed in detail. MSC: 46C05; 49J20; 93C20

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Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction

Cited by 198 publications

References 27 publications

European Vanilla Option Pricing Model of Fractional Order without Singular Kernel

European Vanilla Option Pricing Model of Fractional Order without Singular Kernel

Fractional operator without singular kernel: Applications to linear electrical circuits

Optimality conditions for fractional differential inclusions with nonsingular Mittag–Leffler kernel

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